Q: Evaluate $a_n$ in the Fourier series of the function $f(x) = x + x^2, -\pi < …

By exploiting the fundamental symmetric properties of odd and even functions during integration over $[-\pi, \pi]$, the Fourier cosine coefficient accurately evaluates to $a_n = \frac{4(-1)^n}{n^2}$.

Q: Define Cauchy's $(\epsilon - \delta)$ definition of continuity.

Cauchy's strict mathematical definition states that a function $f(x)$ is continuous at $c$ if, for any $\epsilon > 0$, there exists a $\delta > 0$ such that $|x - c| < \delta$ guarantees $|f(x) - f(c)| < \epsilon$.

Q: Write Euler's theorem on homogeneous function.

Euler's fundamental theorem states that for any continuously differentiable homogeneous function $u(x,y)$ of exact algebraic degree $n$, the strict partial differential relation $x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} = n \cdot u$ holds true.

Q: Evaluate:

By applying the substitution $2x = t$, this improper definite integral perfectly transforms into the standard Gamma function $\Gamma(7)$, rigorously evaluating to the exact rational fraction $\frac{45}{8}$.

Q: Evaluate $\iint xy \, dx \, dy$, where the region of integration is $x + y < …

By carefully setting up the iterated double integral over the triangular planar region bounded by $x+y=1$ in the first quadrant, the exact mathematical volume evaluates to $1/24$.

Q: Change the order of integration of the following double integration:

Reversing the rigid order of integration physically splits the complex triangular region horizontally, resulting in the sum of two distinct mathematical integrals: $\int_0^a \int_0^y f(x,y) \, dx \, dy + \int_a^{2a} \int_0^{2a-y} f(x,y) \, dx \, dy$.

Q: If $\vec{f} = x^2y\hat{i} - 2xy^2z\hat{j} + 3x^2z\hat{k}$, find $\text{div } …

By mathematically computing the partial derivatives of each vector component and evaluating them exactly at the coordinate point $(3, -1, -2)$, the scalar divergence is rigorously determined to be $-3$.

Q: State Stokes theorem.

Stokes' Theorem establishes a profound mathematical equivalence between the macroscopic circulation (line integral) of a vector field around a closed curve and the microscopic rotation (surface integral of the curl) over any bounding surface.

Question 1

Find the limit of the sequence $\langle x_n \rangle$, where

Accepted Answer

By factoring out the highest power of $n$ from both the numerator and the denominator, the limit fundamentally evaluates exactly to the rational fraction $5/7$.

Question 2

Write the power series expansion of logarithm function $\log(1+x)$.

Accepted Answer

The infinite power series expansion for the logarithmic function $\log(1+x)$ is rigorously defined as $x - \frac{x^2}{2} + \frac{x^3}{3} - \dots$, which absolutely converges for $-1 < x \le 1$.

Question 3

Evaluate $a_n$ in the Fourier series of the function $f(x) = x + x^2, -\pi < …

Accepted Answer

By exploiting the fundamental symmetric properties of odd and even functions during integration over $[-\pi, \pi]$, the Fourier cosine coefficient accurately evaluates to $a_n = \frac{4(-1)^n}{n^2}$.

Question 4

Define Cauchy's $(\epsilon - \delta)$ definition of continuity.

Accepted Answer

Cauchy's strict mathematical definition states that a function $f(x)$ is continuous at $c$ if, for any $\epsilon > 0$, there exists a $\delta > 0$ such that $|x - c| < \delta$ guarantees $|f(x) - f(c)| < \epsilon$.

Question 5

Write Euler's theorem on homogeneous function.

Accepted Answer

Euler's fundamental theorem states that for any continuously differentiable homogeneous function $u(x,y)$ of exact algebraic degree $n$, the strict partial differential relation $x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} = n \cdot u$ holds true.

Question 6

Evaluate:

Accepted Answer

By applying the substitution $2x = t$, this improper definite integral perfectly transforms into the standard Gamma function $\Gamma(7)$, rigorously evaluating to the exact rational fraction $\frac{45}{8}$.

Question 7

Evaluate $\iint xy \, dx \, dy$, where the region of integration is $x + y < …

Accepted Answer

By carefully setting up the iterated double integral over the triangular planar region bounded by $x+y=1$ in the first quadrant, the exact mathematical volume evaluates to $1/24$.

Question 8

Change the order of integration of the following double integration:

Accepted Answer

Reversing the rigid order of integration physically splits the complex triangular region horizontally, resulting in the sum of two distinct mathematical integrals: $\int_0^a \int_0^y f(x,y) \, dx \, dy + \int_a^{2a} \int_0^{2a-y} f(x,y) \, dx \, dy$.

Question 9

If $\vec{f} = x^2y\hat{i} - 2xy^2z\hat{j} + 3x^2z\hat{k}$, find $	ext{div } …

Accepted Answer

By mathematically computing the partial derivatives of each vector component and evaluating them exactly at the coordinate point $(3, -1, -2)$, the scalar divergence is rigorously determined to be $-3$.

Question 10

State Stokes theorem.

Accepted Answer

Stokes' Theorem establishes a profound mathematical equivalence between the macroscopic circulation (line integral) of a vector field around a closed curve and the microscopic rotation (surface integral of the curl) over any bounding surface.

Question 11

Prove that:

Accepted Answer

This fundamental mathematical proof establishes the exact algebraic relationship between the Beta and Gamma functions by representing their product as a double integral in Cartesian coordinates, and then elegantly transforming it into polar coordinates to isolate the component functions.

Question 12

Test the convergence of the following series:

Accepted Answer

The convergence of the infinite mathematical series is rigorously tested using Raabe's Test. After determining the limit evaluates exactly to $1/2$ (which is strictly less than 1), the mathematical sequence is unequivocally classified as a divergent series.

Question 13

Find a Fourier series for the function $f(x) = x^2$ in the interval $-\pi < x…

Accepted Answer

By determining the exact Fourier series for the even parabolic function $f(x) = x^2$ and cleverly evaluating the resulting infinite trigonometric series at $x=0$, we successfully deduce the elegant mathematical identity $\frac{\pi^2}{12} = \frac{1}{1^2} - \frac{1}{2^2} + \frac{1}{3^2} - \dots$

Question 14

Find the equations of the tangent plane and normal to the surface $x^3 + y^3 …

Accepted Answer

By rigorously calculating the 3D gradient vector $
abla F$ at the specified coordinate point, we definitively derive the tangent plane equation $x - 3y - 2z + 7 = 0$ and the orthogonal normal line equation $\frac{x-1}{1} = \frac{y-2}{-3} = \frac{z+1}{-2}$.

Question 15

Evaluate the point where the function $x^3y^2(1 - x - y)$ will have maxima. A…

Accepted Answer

By strictly solving the system of first-order partial derivatives and verifying with the second derivative test, the absolute maximum of the function $f(x,y)$ is mathematically proven to occur at $(1/2, 1/3)$ with a value of $1/432$.

Question 16

Evaluate the integral $\int_0^1 \int_0^x \frac{x^3 \, dy \, dx}{\sqrt{x^2+y^2…

Accepted Answer

By radically transforming the complex Cartesian double integral into polar coordinates, the integration becomes mathematically viable, cleanly evaluating step-by-step to the exact logarithmic value $\frac{1}{4} \ln(\sqrt{2} + 1)$.

Question 17

If $\vec{a}$ and $\vec{b}$ are differentiable vector point functions, then pr…

Accepted Answer

This vector calculus identity is rigorously proven by meticulously applying the divergence operator $
abla \cdot$ to the cross product using strict vector product rules and exploiting the invariant cyclic properties of the scalar triple product.

Question 18

Find the volume of spindle shaped solid generated by revolving the Astroid $x…

Accepted Answer

This advanced integral calculus problem strictly determines the exact volumetric capacity of a solid spindle generated by revolving a hypocycloid (astroid) perfectly around the x-axis. Using rigorous parametric substitution and Beta function integration techniques, the final derived volume is mathematically proven to be $\frac{32\pi a^3}{105}$.

Question 19

If $u = \log(x^3 + y^3 + z^3 - 3xyz)$, then prove that:

Accepted Answer

This problem meticulously proves two specific identities related to a complex logarithmic function. By defining the sum of partial derivatives as a differential operator $D$, we systematically compute $D(u) = \frac{3}{x+y+z}$ and rigidly apply the operator a second time to flawlessly derive $D^2(u) = \frac{-9}{(x+y+z)^2}$.

Question 20

Find half range cosine series for the function $f(x) = 2x-1, 0 < x < 1$. Henc…

Accepted Answer

This complex problem requires deriving the half-range Fourier cosine series for the linear function $f(x) = 2x - 1$ strictly over the interval $(0, 1)$. By evaluating the rigorously constructed infinite series at the boundary $x=0$, we mathematically deduce the specific infinite sum identity $\frac{\pi^2}{8} = 1 + \frac{1}{9} + \frac{1}{25} + \dots$

Question 21

Find the volume of the tetrahedron bounded by the coordinate planes and the p…

Accepted Answer

This highly geometric problem elegantly determines the total physical volume of a 3D tetrahedron bounded by Cartesian planes by employing an affine coordinate transformation. This rigorous mapping translates the complex skewed tetrahedron perfectly into a standard unit simplex, resulting in a finalized volume of $\frac{abc}{6}$.

Question 22

State Gauss's divergence theorem. Verify Gauss's divergence theorem for $\vec…

Accepted Answer

This highly rigorous proof verifies the Gauss Divergence Theorem by explicitly calculating both the 3D volume integral of the field's divergence and the cumulative 2D surface integrals of its total flux across all four faces of a unit tetrahedron. Both completely independent methods evaluate precisely to $\frac{1}{8}$.

Mathematics I